Abstract
In this work, we study a Markov model of cyber threats that act on a computer system. Within the framework of the model the computer system is considered as a system with failures and recoveries by analogy with models of reliability theory. To estimate functionally-temporal properties of the system we introduce a parameter called the lifetime of the system and defined as the number of transitions of the corresponding Markov chain until the first hit to the final state. Since this random variable plays an important role at evaluating a security level of the computer system, we investigate in detail its random distribution for the case of mutually exclusive cyber threats; in particular, we derive explicit analytical formulae for numerical characteristics of its distribution: expected value and dispersion. Then we generalize substantially the Markov model dropping the assumption that cyber threats acting on the system are mutually exclusive. This modification leads to an extended Markov chain that has (at least qualitatively) the same structure as the original chain. This fact allowed to generalize the above analytical results for the expected value and dispersion of the lifetime to the case of non-mutually exclusive cyber threats. At the end of the work the Markov model for non-mutually exclusive cyber threats is used to state a problem of finding an optimal configuration of security remedies in a given cyber threat space. It is essential that the formulated optimization problems belong to the class of non-linear discrete (Boolean) programming problems. Finally, we consider an example that illustrate the solution of the problem on selecting the optimal set of security remedies for a computer system.
Highlights
We study a Markov model of cyber threats that act on a computer system
To estimate functionally-temporal properties of the system we introduce a parameter called the lifetime of the system and de ned as the number of transitions of the corresponding Markov chain until the rst hit to the nal state
Since this random variable plays an important role at evaluating a security level of the computer system, we investigate in detail its random distribution for the case of mutually exclusive cyber threats; in particular, we derive explicit analytical formulae for numerical characteristics of its distribution: expected value and dispersion. en we generalize substantially the Markov model dropping the assumption that cyber threats acting on the system are mutually exclusive. is modi cation leads to an extended Markov chain that has the same structure as the original chain. is fact allowed to generalize the above analytical results for the expected value and dispersion of the lifetime to the case of non-mutually exclusive cyber threats
Summary
В настоящем разделе мы напомним основные положения модели киберугроз, предложенной в [14], а также приведем соответствующие аналитические результаты, полученные в наших предыдущих работах [15, 16]. Согласно сделанным предположениям мы можем считать, что в каждый момент времени система находится в одном из состояний 0, 1, ... В случае действия -ой угрозы система переходит в состояние , где = 1, 2, ... Естественно также предположить, что в начальный момент времени = 0 система находится в безопасном состоянии: 0(0) = 1, 1(0) = 2(0) = ⋯ = +1(0) = 0. Выражение (1) представляет собой рекуррентную формулу, выражающую вероятность состояния через вероятности состояний системы в предыдущий момент времени. Мы приведем здесь только вид функции 0( ), так как для дальнейших рассуждений ее будет достаточно:. Что в рамках данного приближения система в нечетные моменты времени практически никогда не обнаруживается в безопасном состоянии, так как в эти моменты времени на систему с большой вероятностью воздействует какая-либо из угроз
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